For Milka Pablo, 35, it was no different. Her 3-year-old daughter, Darly, screamed and tried to wiggle free from her mother’s embrace.

“I want Miss. I want Miss,” Darly cried, calling for the social worker at the shelter where she had been living since mother and daughter were separated by federal agents at the southwestern border.

The tearful reunions — ordered by a court in California — came as the government said that it would release hundreds of migrant families wearing ankle bracelet monitors into the United States, effectively returning to the “catch and release” policy that President Trump promised to eliminate.

Faced with a pair of court orders restricting immigration detentions, federal officials said that they could not hold all of the migrant families who had been apprehended. They said that their hands were tied by dueling requirements to release children from detention after 20 days and also to keep them with their parents or other adult relatives.

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Denis Espinoza, 33, holding his 13-month-old daughter, Carmen, at a bus station in Phoenix on Tuesday. Mr. Espinoza was released from detention and allowed to reunite with her 20 days after their separation.CreditVictor J. Blue for The New York Times

Trump administration officials also said that they have stopped referring migrant adults who enter the United States with children for prosecution.

“Parents with children under the age of 5 are being reunited with their children and then released and enrolled into an alternative detention program,” Matthew Albence, Immigration and Customs Enforcement’s executive associate director of enforcement and removal operations, told reporters on Tuesday.

He said that means the migrants will be given ankle bracelets “and released into the community.”

Government officials said they were struggling to meet Tuesday’s court-ordered deadline to reunite 102 migrant children under 5 with their parents; only about one-third were expected to be reunited by then.

The reunions that did happen were chaotic. Parents were warned that pickup and drop-off times could change throughout the day. The federal agency that oversees the care of migrant children, the Department of Health and Human Services, was still conducting background checks on parents into Tuesday morning.

In Phoenix, the reunions were marked by confusion and heartbreak.

As Ms. Lopez and Ms. Pablo waited at a Greyhound station to board eastbound buses, their children called each other sister and brother. They were yet to utter the word “mami,” Spanish for mother, to the women cuddling, stroking and feeding them. But they were calmer, the mothers said.

Darly, who had been potty-trained before the separation, had regressed to diapers. Ederson bounced up and down on his mother’s lap and downed Doritos with gusto. All of the adults were fitted with ankle monitors.

“I want to go with my little sister,” he said, pointing to a 13-month-old named Carmen in the arms of Denis Espinoza, her Honduran father who was released from detention to recover her 20 days after they were separated.

“See,” said his mother, “he thinks those are his siblings.”

The Justice Department has maintained that its “zero tolerance” immigration policy — which focuses on prosecuting all adults who illegally enter the United States but not necessarily detaining them — is still intact. The department has also said that it is prosecuting all of the cases it receives from immigration agents. Attorney General Jeff Sessions has predicted the tough immigration stance will discourage people from illegally entering the country.

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Ms. Pablo wearing an ankle monitor, with her daughter, Darly.CreditVictor J. Blue for The New York Times

Katie Waldman, a spokeswoman for the Department of Homeland Security, called the return to catch and release unfortunate and said that it would serve “as a pull factor for increased future illegal immigration.”

She added, “D.H.S. will continue to work with Congress to find a solution to this problem.”

Mr. Trump has for years railed against catch and release, blaming it for crimes and violence committed by illegal immigrants during the Obama administration. But the Trump administration has similarly struggled with deterring waves of migrants from Central and South America — and, once they enter the country, processing them through the legal system humanely.

Migrants with children who are apprehended in the United States are now given a notice to appear in court and being told, “Welcome to America,” according to a senior Department of Homeland Security official, who spoke on the condition of anonymity because he was not authorized to be identified by name.

Mr. Albence said the ankle monitor would track the families being released, but that ICE would consider other methods to ensure migrants show up for court. A total of about 80,000 migrants wearing tracking devices currently live in the United States, including migrants who were released from detention before the zero-tolerance policy was enacted in April.

Soon after the policy was announced, images of children in cages and audio of toddlers crying as they were separated from their parents prompted widespread public outrage, including from the Republican Party and Christian conservatives.

In response, on June 20, Mr. Trump issued an executive order that said migrant children could no longer be separated from their adult relatives. That effectively limited detention to 20 days — a timeline set under a 1997 court settlement known as Flores — for migrant adults apprehended with children.

The Trump administration has asked Judge Dolly M. Gee of Federal District Court in Los Angeles to amend the Flores settlement to allow children to be detained for longer periods of time. It also has asked Congress for new laws to override the court order.

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Mirce Alba Lopez, 31, said her 3-year-old son, Ederson, “didn’t recognize me” when the pair were reunited in Phoenix.CreditVictor J. Blue for The New York Times

On Monday, Judge Gee refused to amend the settlement.

“Defendants seek to light a match to the Flores agreement and ask this court to upend the parties’ agreement by judicial fiat,” Judge Gee, who was appointed by President Barack Obama, wrote in her order.

She called the Justice Department’s request to amend the settlement “cynical” and said it was an attempt to “shift responsibility to the judiciary for over 20 years of congressional inaction and ill-considered executive action that have led to the current stalemate.”

The Trump administration is expected to appeal her decision. But for now, it has little choice but to release families with ankle bracelets and hope they will show up for court appearances.

Catch and release is a term with no legal definition and has been used as a pejorative alternative to jailing illegal immigrants. Leon Fresco, a former senior Justice Department immigration lawyer, said putting ankle bracelets on migrants is a return to what the Trump administration itself has described as catch and release; ending its practice was a top priority of the labor union that represents Border Patrol agents and endorsed Mr. Trump’s 2016 candidacy.

The use of ankle bands for migrant families may be short-lived. Judge Dana Sabraw of Federal District Court in San Diego said that he would consider a motion to let migrant families choose between remaining in long-term detention with ICE or voluntarily giving their children to Health and Human Services while they await legal proceedings.

Chris Rickerd, a lawyer with the American Civil Liberties Union, accused the Trump administration of largely manufacturing the current immigration crisis.

“Border residents and people around the country realize it’s this administration making people stand in the sun, waiting to cross into the United States,” Mr. Rickerd said. “It’s this administration separating children from their families. And it’s this administration taking this zero-tolerance policy stance when illegal immigration numbers historically have declined.”

That chaos was highlighted Tuesday by the government’s struggle to comply with the court order in San Diego to reunite children under 5 with their families. Federal officials said that they had reunited four families so far, with another 34 reunions scheduled by day’s end.

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A Guatemalan father and son, who crossed the United States-Mexico border illegally, were apprehended last month by Border Patrol agents in San Diego.CreditJae C. Hong/Associated Press

Additionally, officials gave no indication of whether they would meet a July 26 deadline to reunite all remaining migrant children who had been separated from their parents.

Chris Meekins, a senior Health and Human Services Department official, pointed to safety concerns to explain the delay and insisted that the reunifications could not be rushed.

“Our process may not be as quick as some might like, but there is no question that it is protecting children,” Mr. Meekins said in a conference call with reporters.

In some cases, he said, “If we had just reunited kids with the adults, we would be putting them in the care of a rapist, a kidnapper, a child abuser and someone who was charged with murder in their home nation.”

One of the biggest operators of migrant-youth shelters in the United States, Southwest Key Programs, said its staff had dispatched several children on Tuesday from its shelters to return to their parents.

“Our staff came in early, made sure every backpack was full and every child got a hug and a goodbye,” Juan Sánchez, the nonprofit group’s president and chief executive, said in a statement. “And the kids hugged us back. They were excited to be on their way to be with their families. And we were thrilled for them.”

The nonprofit declined to discuss how many children under 5 were released.

The reunification process has highlighted how traumatic separations stemming from the zero-tolerance policy have been.

One Honduran father had been warned by border agents that his child would be taken away, and he had been given an opportunity to explain to his son what would happen. He beamed while his son asked questions and played with toys when they were reunited Tuesday at a federal immigration office in Michigan.

A second father was not given a chance to tell his 3-year-old son that they would be separated, according to Abril Valdes, a lawyer for both men. His child had stopped talking soon after being taken into government custody. The father and son cried throughout the reunification meeting. The boy said little and refused to take any toys or leave his father’s arms.

“I feel like he’s still in shell shock,” said Ms. Valdes, their lawyer.

Correction:

An earlier version of this article misspelled the surname of a lawyer with the American Civil Liberties Union. It is Rickerd, not Rickard.

Miriam Jordan reported from Phoenix, Katie Benner and Ron Nixon from Washington, and Caitlin Dickerson from New York. Manny Fernandez contributed reporting from Houston, Maggie Haberman from New York, and Kirk Semple from Guatemala City.